Conventional traveling wave oscillators, employed in many electronics applications requiring high-speed clock signals (e.g., on the order of several gigahertz), are linear systems that generally employ a linear transmission line over which sinusoidal waves of a particular frequency are propagated. In such systems, linear amplifiers are employed to overcome resistive losses present in the transmission line so as to maintain the signal strength of the traveling sinusoidal wave and permit oscillation. While such linear oscillator systems have found application in the area of high-speed electronics, for example, ongoing research in nonlinear systems has raised interest in the practical merits of such systems as an alternative to linear systems, and possible applications for non-sinusoidal waveforms traveling in nonlinear media.
It is generally appreciated in mathematics and the physical sciences that any arbitrary non-sinusoidal waveform in space and/or time may be represented by multiple sinusoidal waveforms of varying frequencies, amplitudes and phases superimposed upon one another. This idea of a given waveform having multiple different-frequency sinusoidal components, commonly referred to as a Fourier spectrum, has interesting implications for non-sinusoidal waveforms that travel over some distance from one point to another via some medium.
In particular, an arbitrary waveform traveling through a medium may or may not be significantly affected or altered (e.g., distorted in shape) by the medium, depending on how the waveform's constituent sinusoidal components are affected by the medium. The medium of a perfect vacuum may be viewed as a sort of reference point for waveform propagation; in the medium of a perfect vacuum, often referred to as “free space,” it is generally presumed that an arbitrary waveform is not significantly affected while traveling through free space, as there is theoretically no material in the medium of the vacuum to somehow affect or impede the propagation of the sinusoidal components making up the waveform. Accordingly, when sinusoidal waves travel through a medium other than free space, they propagate more slowly than they would in a theoretical vacuum, as the material making up the medium through which the waves travel somehow affects or impedes the propagation of the waves.
The speed of a given sinusoidal component traveling through some medium is commonly referred to as the “phase velocity” of the component (often denoted as vp), and is generally related to various physical properties of the material that makes up the medium. For each sinusoidal component of a waveform, the phase velocity of that component is given as vp=ƒλ, where f is the frequency of the component and λ is the wavelength of the component.
Various media that are encountered in practical applications involving waveform propagation may be generally characterized by two noteworthy properties of the medium that may affect waveform propagation. One of these properties is conventionally referred to as “dispersion” and another of these properties is conventionally referred to as “nonlinearity.” In a medium with significant dispersive properties, the frequency of a given sinusoidal waveform may affect the speed with which it travels through the medium. On the other hand, in a medium with significant nonlinear properties, the velocity of the waveform differs for each point on the waveform, usually depending on the amplitude of the point. In general, various media may possess some degree of both dispersive and nonlinear properties; in practical applications, one or both of these properties may or may not present a significant issue with respect to waveform distortion.
For example, if a medium does not have significant dispersive properties for a given application (i.e., a “non-dispersive” medium for a particular frequency range), essentially all sinusoidal waves having frequencies within a particular range travel through the medium at essentially the same speed. Thus, in a non-dispersive medium, an arbitrary waveform traveling through the medium theoretically maintains its shape, as all of the constituent sinusoidal components making up the waveform travel together at the same speed. Again, one common example of a non-dispersive medium is a theoretical vacuum or “free space,” as discussed above.
In contrast to a non-dispersive medium, a dispersive medium is one in which the properties of the medium that determine phase velocity vp depend on frequency ƒ. Stated differently, the phase velocity of a given frequency component is a function of the frequency of that component and, as a result, different frequencies travel at different speeds through the dispersive medium. This situation may significantly affect an arbitrary waveform traveling through the dispersive medium; namely, the different sinusoidal frequency components making up the waveform travel at respectively different speeds and “spread out” from one another, which inevitably leads to distortion (e.g., broadening) of the waveform. This condition is referred to as “dispersion,” and is observed to various extents in many real-world situations involving waveforms traveling through media. Some common examples of dispersion include the spreading out of waves created by a drop of water onto a pond, as well as waveform distortion in electronic or optical communications (in which electrical signals travel significant distances through conductive or semiconductive material, or light travels significant distances through optical fibers).
As discussed above, not only may the frequency of a given sinusoidal component affect the speed with which it travels through a medium, but in some media the amplitude or “strength” of the component also may affect the speed with which each point on a waveform travels. One example of a nonlinear medium is given by waves in shallow water (e.g., consider the ocean near a beach). As the waves approach the beach they are initially symmetric. As they get closer to the shoreline, however, the peaks travel faster than the troughs due to the nonlinearity in water, thus causing the front face to steepen. Eventually this slope becomes too great for the wave and it breaks.
While either dispersion or nonlinearity each can lead to significant distortion of an arbitrary waveform propagating in a given medium, in some instances a medium may have both dispersive and nonlinear qualities, i.e., both the frequency and amplitude of a waveform may affect the speed with which it travels through the medium. In one sense, this may be viewed as a somewhat chaotic circumstance for an arbitrary waveform traveling through the medium; more specifically, it should be appreciated that an arbitrary waveform propagating through a dispersive and nonlinear medium very likely would quickly distort and ultimately “disappear,” as its constituent sinusoidal components seemingly travel randomly at different speeds based on their respective frequencies and/or amplitudes.
However, if certain conditions are fulfilled, namely, if there is a very particular balance between the effects of dispersion (which tend to spread out or broaden a waveform) versus the effects of nonlinearity (which in some instances tends to “bunch up” or squeeze together different frequency components of a waveform), the unexpected may happen; a particularly shaped waveform may be excited in the dispersive nonlinear medium and propagate through the medium without distortion. Such a remarkable waveform that exploits a precise dynamic balance between dispersion and nonlinearity to maintain its shape is called a “soliton.”
A soliton waveform refers to a bell-shaped pulse that propagates by itself in a dispersive nonlinear medium without changing its shape. This solitary wave also can survive collisions with other solitary waves without altering its speed or shape, in a manner similar to that of classical particles such as electrons and protons; hence the name “soliton,” i.e., a solitary wave which behaves like a classical particle (e.g., electron or proton). The first observation of a soliton was made by John Scott Russel in 1838 when he noted an unusually persistent water wave traveling an appreciable distance down a barge canal without changing its shape. Many years later, Russel's observation was given a theoretical framework by the Dutch mathematicians D. J. Korteweg and G. de Vries, who derived arguably the most famous of all nonlinear equations (referred to as the KdV equation) to describe the interaction of dispersion and nonlinearity that gives rise to the existence of a soliton waveform.
In particular, the KdV equation describes the behavior of a waveform in space and time in the presence of dispersion and nonlinearity. If the shape or profile of the waveform (i.e., its “height”) at a given point z in space, and at a given time t, is denoted as u (z, t), a general form for the KdV equation is given by:
                                                        ∂              u                                      ∂              t                                +                      6            ⁢            u            ⁢                                          ∂                u                                            ∂                z                                              +                                                    ∂                3                            ⁢              u                                      ∂                              z                3                                                    =        0.                            (        1        )            
The first term of Eq. (1) signifies that it is a first order evolution equation in time (i.e., a determination of what happens to the overall profile or shape of the waveform over time). The second term of Eq. (1) accounts for nonlinearity and is amplitude dependent (as evidenced by the factor u multiplying the first order partial derivative with respect to space z), while the third term accounts for dispersion. Arbitrary coefficients for these terms can be arranged by resealing the dependent and independent variables.
The soliton solution to Eq. (1) is given by:
                                          u            ⁡                          (                              z                ,                t                            )                                =                      A            ⁢                                                  ⁢                          sech              2                        ⁢                                          A                2                                      ⁢                          (                              z                -                                  2                  ⁢                                                                          ⁢                  At                                            )                                      ,                            (        2        )            which is a smooth bell-shaped pulse that completely depends on the parameter A (which represents the maximum amplitude of the pulse). Again, Eq. (2) represents the evolution of a snapshot of a soliton waveform taken at one time to another snapshot at some later time (it should be appreciated that a valid description also may be maintained by evolving a function of time in the spatial variable z, in which case the roles of the independent variable z and t in Eq. (1) are reversed).
Eq. (2) describes a family of soliton solutions, all determined by the parameter A. FIG. 1 shows two examples of solitons described by Eq. (2). As the nonlinearity in the medium of propagation depends on the amplitude of the soliton waveform, and the dispersion in the medium depends on the width (i.e., the frequency content) of the soliton waveform, it should be appreciated that more than one combination of width and amplitude can satisfy the balance needed for a soliton; hence, taller solitons are relatively narrow (larger A), whereas as shorter solitons are relatively broader (smaller A), as illustrated in FIG. 1. Additionally, it may be noted from Eq. (2) that the speed of a soliton is dependent on the pulse amplitude A, such that taller solitons (larger A) travel more quickly than shorter solitons. In sum, the height, width and speed of KdV solitons are all interrelated, in that a soliton of a specific amplitude has a specific width and travels at a specific speed.
In the physical sciences, solitons have generated significant interest for practical applications in the areas of hydrodynamics, chemistry, plasma physics, optical communications, information encoding and decoding, pulsed power technology, spectroscopy, and neural networks. Generally speaking, any application that may utilize or require sharp reproducible pulses may implicate the generation and propagation of solitons.
In addition to solitons, the KdV equation has a periodic solution, often referred to as “cnoidal” waves, which essentially are constructed from a nonlinear sum of translated soliton solutions. Accordingly, in one sense, a cnoidal wave may be viewed figuratively as a train of soliton pulses all propagating at the same speed at some fixed distance from one another.
In practical applications involving soliton (or cnoidal wave) propagation through some medium, features other than dispersion and nonlinearity often must be considered. Perhaps the most common is attenuation, or loss of signal power as the soliton propagates in the medium, due to dissipative (resistive) properties of the medium. These dissipative properties may be linear or nonlinear, as well as frequency dependent or independent. Also, perturbations to the system or inhomogeneities in the system in which a soliton propagates also may need to be accounted for. Accordingly, a general modification to the KdV equation may be made to account for features such as attenuation, perturbation and/or inhomogeneity by adding a term P(u) to the right hand side of Eq. (1):
                    ∂        u                    ∂        t              +          u      ⁢                        ∂          u                          ∂          z                      +                            ∂          3                ⁢        u                    ∂                  z          3                      =            P      ⁡              (        u        )              .  
Generally speaking, the evolution of an initial condition (e.g., an arbitrary waveform, perturbation, disturbance, noise, etc.) in a dispersive nonlinear system described by the KdV equation can be broken down into two categories, namely, parts which develop into solitons as described by Eq. (2) above, and those which do not (often referred to as a dispersive wave or radiation). In many applications, the dispersive wave or radiation can be treated as a transient, because it eventually dies away, while the soliton(s) remain intact. It is a remarkable feature of the KdV equation that almost every reasonably shaped initial condition develops into one or more solitons; in this sense, solitons have a surprisingly natural stability and robustness to noise in the face of chaotic conditions presented by a nonlinear system. If an initial condition is such that multiple solitons evolve, they will eventually separate because, as discussed above, solitons of different amplitudes travel at different speeds (i.e., higher amplitude solitons travel relatively faster).
Solitons (or cnoidal waves), or nonlinear pulses that closely resemble pure solitons (i.e., “quasi-soliton” pulses) are known to propagate in the medium of a nonlinear transmission line, which may be described by the KdV equation. For the following discussions, the terms “soliton” or “solitons” refer not only to pure soliton waveforms or cnoidal waves, but also are intended to encompass nonlinear pulses that may not have exactly a soliton waveform, but significantly resemble solitons and behave effectively like solitons in a given environment.
Many experiments on soliton propagation have been conducted using nonlinear transmission lines constructed from lumped components and implemented as a “lattice,” an example of which is illustrated in FIG. 2. The nonlinear transmission line lattice 30 shown in FIG. 2 includes a number of inductors 32 each having an inductance L and a number of varactor diodes 34 arranged in a reverse-biased configuration to provide a voltage-dependent capacitance C(V); in particular, the capacitance of the varactor diodes 34 increases as the voltage V across the diodes decreases, according to a nonlinear relationship which may be approximated by:
                                          C            ⁡                          (              V              )                                =                                    Q              0                                                      F                o                            -              V              -                              V                0                                                    ,                            (        3        )            where Fo and Qo are parameters that describe the nonlinearity and Vo is a reverse DC bias voltage applied across the varactor diodes, in addition to the signal voltage V, to provide for minor adjustments in the nonlinear response of the diodes. Hence, the nonlinearity of the lattice 30 is attributed to the nonlinear voltage-dependent capacitance C(V) provided by reverse-biased varactor diodes 34, whereas the overall lattice-like arrangement of lumped components (in essence acting like a multiple-order filter) provides the dispersion to balance the nonlinearity in the case of a soliton waveform.
With the appropriate nonlinear characteristic, the lumped nonlinear transmission line is equivalent to what is conventionally known in particle physics as a “Toda Lattice,” which models nonlinear potentials between particles. The Toda Lattice may be closely approximated by the KdV equation and thus has soliton and cnoidal solutions. For this reason, the lumped nonlinear transmission line has received considerable attention for studying solitons.
The inductors and varactor diodes in the lattice of FIG. 2 are illustrated as “ideal” inductive and capacitive components. In practical implementations, however, the physical structure of these components results in non-ideal behavior that must be appropriately modeled. For example, a realistic model for the inductors 32 must include the effects of a series-resistance of the inductor winding, as well as inter-winding capacitance and high frequency parallel resistive losses. In particular, the inductor winding resistance is generally considered to most significantly contribute to signal power loss on the nonlinear transmission line (i.e., signal attenuation), whereas inter-winding capacitance contributes to dispersion. Likewise, non-ideal behavior in the varactor diodes 34 may be appropriately modeled by considering a series-resistance with the diode, which further contributes to high frequency losses and overall signal attenuation with propagation along the lattice 30.
In some early research regarding soliton propagation on a nonlinear transmission line, a loop of nonlinear transmission line was employed to increase the apparent length of the transmission line so as to analyze soliton propagation over significant distances (e.g., see Michel Remoissenet, “Waves Called Solitons, Concepts and Experiments,” Third Edition, Springer-Verlag, 1999, ISBN 3-540-65919-6, hereby incorporated herein by reference). FIG. 3 illustrates a diagram of such an apparatus, employing two switches 36, a pulse generator/switch driver 38, and a nonlinear transmission line 30. For simplicity, the varactor diodes 34 of the transmission line are represented in FIG. 3 as variable capacitors. In the arrangement of FIG. 3, the switches 36 first are placed in their upper positions, such that one end of the nonlinear transmission line is coupled to an output of the pulse generator 38, while another end of the transmission line is terminated with a resistance RL. The pulse generator outputs a pulse having an arbitrary waveform which, due to a balance between dispersion and nonlinearity presented by the nonlinear transmission line, evolves into a soliton. If the switches are then placed in their lower positions, the opposite ends of the transmission line are coupled together to form a closed loop, which permits the soliton to circulate in the loop (as if traveling along a significant length of nonlinear transmission line). Due to signal attenuation resulting from the resistive losses inherent in the nonlinear transmission line, the amplitude of the circulating soliton gradually decreases (while the width of the soliton correspondingly increases), until eventually the soliton dies out.
In subsequent research, in an attempt to overcome losses in the medium, an amplifier was employed with a loop of nonlinear transmission line to implement a soliton oscillator. The addition of the amplifier permitted a theoretically self-sustaining device in which a soliton could conceivably circulate endlessly in the loop of nonlinear transmission line (e.g., see Ballantyne et al., “Periodic Solutions of Toda Lattice in Loop Nonlinear Transmission Line,” Electronics Letters, Vol. 29, No. 7, Apr. 1, 1993, pp. 607-609 and Ballantyne, Gary J., “Periodically Amplified Soliton Systems,” Doctoral Thesis in Electrical and Electronic Engineering at the University of Canterbury, Christchurch, New Zealand, both of which disclosures are incorporated herein by reference). Such an oscillator based on a nonlinear transmission line may be effectively modeled as a Toda Lattice, as discussed above. In such a device, there are four mechanisms operating to produce a soliton oscillation, namely, dispersion, nonlinearity, attenuation and amplification. A steady state is reached when the circuit achieves a balance amongst all four of these mechanisms.
FIG. 4 illustrates such an arrangement, in which a section of nonlinear transmission line 30A (including the inductors 32 and varactor diodes 34) is “tapped” and coupled via a first buffer 58A to a high pass filter 50, and then via a second buffer 58B to a conventional linear amplifier 52. A power supply 56 provides a DC bias voltage for the circuit, so as to appropriately reverse-bias the varactor diodes 34 at some predetermined operating point along a characteristic capacitance/voltage curve for these devices (e.g., see the parameter Vo of Eq. (3) above). The output of the linear amplifier 52 drives the section of nonlinear transmission line 30A to complete the loop. In the desired operation mode, the gain of the linear amplifier is carefully adjusted (via variable resistor 60) to precisely compensate for losses in the circuit, so as to create the appropriate conditions for soliton propagation in the loop.
FIG. 5 illustrates three examples of waveforms observed in the oscillator, each representing a soliton circulating repeatedly around the loop given a particular amplifier gain setting. As discussed further below, for some particular range of amplifier gains, a relatively lower gain corresponds to a lower pulse amplitude (e.g., uppermost plot in FIG. 5), whereas a relatively higher gain corresponds to a higher pulse amplitude (e.g., lowermost plot in FIG. 5). An output of the oscillator circuit may be taken at the point 64 indicated in FIG. 4 (at the output of the first buffer 58A), although theoretically the circulating soliton pulses may be observed at any point in the circuit. It should be appreciated that in the circuit of FIG. 4, the first buffer 58A provides required isolation between the high pass filter 50 and the line 30 (so that the filter does not unduly load the line), whereas the second buffer 58B provides required isolation between the high pass filter 50 and the amplifier 52 (so that the gain adjustment resistors of the amplifier do not unduly load the high pass filter).
In the circuit of FIG. 4, so as to prevent reflections and maintain unidirectional operation of the oscillator, a termination 54 is employed on the nonlinear transmission line to approximate a matched load for the line. An exact match is not practically possible because the transmission line is nonlinear and dispersive and does not have a constant characteristic impedance. In particular, in the case of a linear dispersionless transmission line, the line could be perfectly terminated with its characteristic impedance
            Z      o        =                  L        C              ,so that each frequency is perfectly absorbed by the resistive termination. In the case of a lumped “LC” transmission line, dispersion dictates that this can only be achieved at one frequency. If the transmission line further is nonlinear, as in the example of FIG. 4, the termination theoretically also should be nonlinear, which is difficult to practically implement.
Accordingly, as a reasonable approximation, a nominal characteristic impedance ζo is selected for the termination 54 based on the nominal capacitance of the varactor diodes at zero volts
      (                  i        .        e        .            ,                        ζ          o                =                              L                          C              ⁡                              (                0                )                                                          )    .A small blocking capacitor Cb is also employed to prevent a voltage divider forming for DC voltages between the transmission line series resistance and the nominal termination resistance ζo. Hence, the relatively simple termination 54 shown in FIG. 4 cannot provide a perfect match, but is nonetheless sufficient at allowing some soliton oscillations to exist on the loop for an observable period of time.
The filter 50 of the circuit shown in FIG. 4 is a simple high-pass filter to inhibit low frequency instability. In particular, because of the positive feedback arrangement of the loop provided by the non-inverting linear amplifier 52 (discussed further below), the filter is required to block the DC bias voltage provided across the nonlinear transmission line by the power supply 56. Without the filter 50, the oscillator sits at the upper DC supply rail of the amplifier when the gain is increased beyond some critical value. Beyond this requirement, the filter cut-off frequency is arbitrarily selected to be low enough so as not to hinder operation of the oscillator (i.e., so as to pass the frequency content of signals that may be supported by the oscillator). Generally speaking, the frequencies supported by the oscillator circuit of FIG. 4 (i.e., the pulse repetition rate, or distance between consecutive soliton pulses) are determined by the inductance values L, range of values for the variable capacitances C(V) and the number of nodes or sections of the nonlinear transmission line 30A over which the soliton travels before the line is tapped and fed back to the high-pass filter 50.
As discussed above, the role of the linear amplifier 52 in the circuit of FIG. 4 is to compensate for transmission line losses and overcome attenuation of a soliton as it propagates around the circuit loop. As can be readily observed in FIG. 4, the amplifier is implemented as a conventional non-inverting linear amplifier whose gain is manually adjustable via the variable resistor 60. Soliton oscillation is initiated by increasing the gain to some critical value which is sufficient to overcome losses in the circuit. Beyond this critical gain, one or more solitons begin circulating in the loop. This implies that a noisy initial state is fashioned into one or more steady state waveforms after successive passes around the loop. In essence, the nonlinear transmission line recursively fashions noise into a shape that can traverse the circuit and be re-amplified back to a sustainable form—in particular, soliton solutions that closely resemble those of the KdV equation discussed above.
Just beyond the critical gain to begin oscillation, further adjustments to the gain of the linear amplifier 52 affect the amplitude of a given soliton generated in the circuit of FIG. 4, as discussed above. In particular, a small gain just above the critical gain produces a relatively smaller amplitude soliton, whereas a larger gain produces a relatively larger amplitude soliton (recall that Eq. (2) above represents a family of solutions with different amplitudes A, wherein the width of the soliton pulse decreases with increasing amplitude and larger amplitude pulses travel faster than smaller amplitude pulses). FIG. 5 illustrates the effect of varying the amplifier gain in the circuit of FIG. 4 just above the critical gain, wherein the uppermost plot represents relatively smaller gain and the lowermost plot represents relatively larger gain. It may be readily observed in FIG. 5 that the soliton width decreases with increasing gain, as expected based on Eq. (2) above.
One problem with the linear amplification scheme in the circuit of FIG. 4 is that the circuit is extremely sensitive to gain adjustments; in particular, the change in gain necessary to overcome losses and generate some steady state soliton oscillation is a few percent at most, and more typically less than one percent. This situation makes it extremely difficult to manually adjust the amplifier gain to precisely those values that permit a desired oscillation. For purposes of illustration, this type of gain adjustment is somewhat like trying to balance a ball on the tip of a pencil, in which a small deviation from the precise balance of forces (e.g., a small disturbance) causes the ball to fall.
In an attempt to lower the sensitivity of amplifier gain adjustment, additional resistive losses are deliberately introduced in the circuit of FIG. 4, in the form of resistors 62 disposed in both the series and shunt branches of the nonlinear transmission line 30A. In the foregoing illustration of a ball balanced on a pencil tip, the deliberate introduction of additional resistive losses is akin to dulling the point of the pencil on which the ball is balanced. These additional losses intentionally introduced in the nonlinear transmission line are required to allow any practical adjustment of the amplifier gain in the circuit of FIG. 4. At the same time, the resistors 62 in the nonlinear transmission line 30A constitute a noteworthy difference from the conventional nonlinear transmission line 30 illustrated in FIGS. 2 and 3, in which losses in the line are attributed primarily to the naturally occurring non-ideal resistive properties of the inductor and varactor diode components. Hence, the addition of the resistors 62 is somewhat of a peculiar departure from the conventional nonlinear transmission line configuration, inasmuch as the line must be made more lossy to facilitate proper operation of the circuit.
It should be appreciated, however, that notwithstanding the intentional addition of extra losses in the circuit, the manual amplifier gain adjustment in the circuit of FIG. 4 is still significantly sensitive. Specifically, in the exemplary waveforms shown in FIG. 5, the amplifier gains that generate the three waveforms are, from top to bottom, 1.092, 1.100 and 1.109, respectively, demonstrating appreciably small differences in gain of less than one percent for corresponding significant differences (e.g., approximately doubling) in pulse amplitude. Again, this situation makes it extremely difficult to manually adjust the amplifier gain to permit a desired oscillation.
Another problematic issue in the circuit of FIG. 4 is that when the gain is further increased beyond the critical value to initiate soliton oscillation, multiple solitons having different amplitudes may be generated. Again, due to the sensitivity of the amplifier gain adjustment, this condition is easily achieved whether or not it is desired; in particular, even if the gain is carefully adjusted initially to permit oscillation of a single soliton having a particular amplitude, system conditions (e.g., filter cutoff frequency) may be such that an external perturbation (e.g., noise) begins circulating in the circuit and is amplified to create one or more additional solitons.
Because of the different amplitudes of the multiple solitons thusly generated, each soliton travels at a different speed and multiple solitons eventually collide with one another in the oscillator circuit of FIG. 4. As discussed above, an interesting characteristic of solitons is that they survive collisions without altering their speed or shape; hence the multiple different-amplitude solitons continue to circulate and collide periodically. FIG. 6 illustrates a collision 66 of two solitons, as a larger amplitude pulse 68 overtakes and collides with a smaller amplitude pulse 70. It should be appreciated that for applications in which a stable and predictable oscillator is desired, collisions of multiple solitons may represent a notably undesirable condition.
In sum, the circuit of FIG. 4 is characterized by a significantly sensitive manual amplifier gain adjustment and somewhat unpredictable multiple-mode operation in which soliton collisions may readily occur. Hence, although the circuit of FIG. 4 provides an interesting laboratory curiosity for research purposes, this type of soliton oscillator implementation is arguably not well-suited for practical applications that require stable single-mode oscillation.